Neurosignalling Summary - Lecture 1 PDF

### Lecture 1 ### Comparing Electronic Circuits and Neurons Concept Electronic Circuit Neurons -------------------- -------------------- ------------------------ Isolation Vacuum Lipid Membrane Resistor/Conductor Cables, Metals Solution, Ion...

### Lecture 1 ### Comparing Electronic Circuits and Neurons Concept Electronic Circuit Neurons -------------------- -------------------- ------------------------ Isolation Vacuum Lipid Membrane Resistor/Conductor Cables, Metals Solution, Ion Channels Charge Carrier Electrons Ions ### Current and Kirchhoff's Laws - **Current (I)**: Defined as the flow of positive charge (+) from positive to negative (or equivalently the flow of negative charge (--) in the opposite direction). The sign of the current indicates its direction. - **Kirchhoff's Current Law**: The net charge in a region remains constant, so the sum of currents flowing into a node equals the sum of currents flowing out of the node, with no loss of charges (I~3~ = I~1~ + I~2~). - **Kirchhoff's Voltage Law**: The directed sum of potential differences (voltages) around a closed loop is equal to zero. ### Electric Potential and Membrane Potential - **Electric Potential (V)**: The potential difference between two points (continuous quantity varying in space). - **Membrane Potential**: The difference between the potential inside the membrane and outside the membrane, i.e., [*V*~in~ − *V*~out~]{.math.inline}. ### Electric Force and Field - A charged particle experiences a force ([*F*]{.math.inline}) in an electric field ([*E*]{.math.inline}): \ [*F* = *q* ⋅ *E*]{.math.display}\ - Where [*q*]{.math.inline} is the charge, and [*E*]{.math.inline} is the strength of the electric field. - The electric field ([*E*]{.math.inline}) is the gradient of the electric potential: \ [\$\$E = - \\frac{\\text{dV}}{\\text{dx}}\$\$]{.math.display}\ ### Space-Charge Neutrality and Membrane Compensation - **Space-Charge Neutrality**: In any given volume, the total charge of cations approximately equals the total charge of anions. When excess negative charges are present inside the cell membrane, they are compensated by positive charges outside the membrane. ### Capacitance and Charge on the Membrane The amount of charge ([*Q*]{.math.inline}) required to establish a membrane potential ([ΔV]{.math.inline}) over a surface with capacitance [*C*]{.math.inline} is: \ [*Q* = *C* ⋅ ΔV]{.math.display}\ ### Faraday's Constant and Ion Quantities - **Faraday's Constant (F)**: The charge of a mole of ions, [*F* = 96485 *C*/*mol*]{.math.inline}. - To calculate the number of moles of ions: \ [\$\$\\mathrm{\\text{Moles\~of\~ions}} = \\frac{Q}{F}\$\$]{.math.display}\ - To calculate the number of ions: \ [\$\$\\mathrm{\\text{Number\~of\~ions}} = \\frac{Q}{F} \\cdot N\_{A}\$\$]{.math.display}\ - Where [*N*~*A*~]{.math.inline} is Avogadro's number (6.022 × 10²³). - The number of ions required to bring a cell from 0 mV to --80 mV is much smaller than the total number of ions present in a cell. Over short timescales (e.g., during a single action potential), ion concentrations do not change significantly. - Excess negative charges inside impermeable membrane are compensated by positive charges outside the membrane. ### Ohm's Law for Drift The drift flux ([*J*~drift~]{.math.inline}) is the flow of particles due to the electric field: \ [*J*~drift~ = *σ* ⋅ *E*]{.math.display}\ Where [*σ*]{.math.inline} is the electrical conductivity, and [*E*]{.math.inline} is the electric field: \ [\$\$E = - \\frac{\\text{dV}}{\\text{dx}}\$\$]{.math.display}\ Thus, \ [\$\$J\_{\\mathrm{\\text{drift}}} = - \\mu \\cdot z \\cdot \\lbrack C\\rbrack \\cdot \\frac{\\text{dV}}{\\text{dx}}\$\$]{.math.display}\ Where: - [*μ*]{.math.inline} is the mobility, - [*z*]{.math.inline} is the valence of the ion, - [\[*C*\]]{.math.inline} is the concentration of ions. Double concentration of ions \[C\] -\> double drift (Jdrift) Double the amount of charge/valence (z) --\> double drift Positive valence /ion charge (z) -\> negative drift (from positive towards negatively charged side) Negative valence/ion charge (z) -\> positive drift (from negative towards positively charged side) ### Fick's Law for Diffusion The diffusion flux ([*J*~diff~]{.math.inline}) is the flow of particles due to a concentration gradient: \ [\$\$J\_{\\mathrm{\\text{diff}}} = - D \\cdot \\frac{d\\lbrack C\\rbrack}{\\text{dx}}\$\$]{.math.display}\ Where [*D*]{.math.inline} is the diffusion coefficient and [\[*C*\]]{.math.inline} is the concentration of ions. ### Einstein Relation Between Diffusion and Mobility The diffusion coefficient [*D*]{.math.inline} is related to mobility [*μ*]{.math.inline} via the Einstein relation: \ [\$\$D = \\text{μ\\ } \\cdot \\ \\frac{k \\cdot T}{q}\$\$]{.math.display}\ Or, \ [\$\$D = \\text{μ\\ } \\cdot \\frac{R \\cdot T}{F}\$\$]{.math.display}\ Where: - [*k*]{.math.inline} is Boltzmann's constant, - [*T*]{.math.inline} is the absolute temperature in Kelvin, - [*q*]{.math.inline} is the charge of a molecule, - [*R*]{.math.inline} is the universal gas constant. ### Nernst-Planck Equation The total flux ([*J*~total~]{.math.inline}) is the sum of the drift and diffusion fluxes: \ [*J*~total~ = *J*~drift~ + *J*~diff~]{.math.display}\ Thus, \ [\$\$J\_{\\mathrm{\\text{total}}} = - u \\cdot z \\cdot \\left\\lbrack C \\right\\rbrack \\cdot \\frac{\\text{dV}}{\\text{dx}} - u \\cdot \\frac{\\text{RT}}{F} \\cdot \\frac{d\\lbrack C\\rbrack}{\\text{dx}}\$\$]{.math.display}\ Which is: \ [\$\$J\_{\\mathrm{\\text{total}}} = - \\left( u \\cdot z \\cdot \\lbrack C\\rbrack \\cdot \\frac{\\text{dV}}{\\text{dx}} + u \\cdot \\frac{\\text{RT}}{F} \\cdot \\frac{d\\lbrack C\\rbrack}{\\text{dx}} \\right)\$\$]{.math.display}\ The current ([*I*]{.math.inline}) per unit cross-section area (in A/cm[^2^]{.math.inline}) is related to the total particle flux ([*J*~total~]{.math.inline}) by: \ [*I* = *J*~total~ ⋅ *z* ⋅ *F*]{.math.display}\ Thus, the expression for current becomes: \ [\$\$I = - \\left( u \\cdot z\^{2} \\cdot F\\ \\cdot \\lbrack C\\rbrack \\cdot \\frac{\\text{dV}}{\\text{dx}} + u \\cdot z \\cdot R \\cdot T \\cdot \\frac{d\\lbrack C\\rbrack}{\\text{dx}} \\right)\$\$]{.math.display}\ This represents the flux of particles due to both electric and chemical forces (ion charge through a surface per time step). ### Nernst Equation (At Equilibrium) When there is no net flux ([*I* = 0]{.math.inline}), the **Nernst equation** defines the equilibrium potential of ions across the membrane. \ [\$\$E\_{x} = V\_{\\text{in}} - V\_{\\text{out}} = \\frac{\\text{RT}}{\\text{zF}} \\cdot ln\\left( \\frac{\\lbrack x\\rbrack\_{\\text{out}}}{\\lbrack x\\rbrack\_{\\text{in}}} \\right)\$\$]{.math.display}\ ### Ion Permeability and Donnan Equilibrium - Most cell membranes at rest are permeable to potassium (K[^+^]{.math.inline}) and chloride (Cl[^−^]{.math.inline}), but much less permeable to sodium (Na[^+^]{.math.inline}) and calcium (Ca[^2+^]{.math.inline}). - The **Donnan Rule of Equilibrium**: Without active transport, the membrane potential should equal the equilibrium potentials of all permeable ions. ### Lecture 2 Membrane Permeability and Flux Equation Membrane permeability defines the flux of ions across a membrane based on the concentration gradient. The equation is: \ [*J* =  − PΔ\[*C*\] =  − *P* × (\[*X*\]~in~−\[*X*\]~out~)]{.math.display}\ Where: - [*J*]{.math.inline} is the flux (rate of ion movement across the membrane). - [*P*]{.math.inline} is the permeability of the membrane. - [*Δ*\[*C*\] = \[*X*\]~in~ − \[*X*\]~out~]{.math.inline} is the concentration difference of the ion across the membrane. This equation can also be derived from **Fick's Law** for diffusion: \ [\$\$J = - D\\frac{d\\lbrack C\\rbrack}{\\text{dx}}\$\$]{.math.display}\ Where: - [*D*]{.math.inline} is the diffusion coefficient. - [\$\\frac{d\\lbrack C\\rbrack}{\\text{dx}}\$]{.math.inline} is the concentration gradient over distance [*x*]{.math.inline}. Given [*J* =  − PΔ\[*C*\]]{.math.inline}, permeability ([*P*]{.math.inline}) can be related to the mobility ([*u*]{.math.inline}) of ions: \ [\$\$P = \\frac{D\\beta}{\\text{Δx}} = \\frac{u \\cdot R \\cdot T}{F}\\frac{\\beta}{l}\$\$]{.math.display}\ Where: - [*u*]{.math.inline} is the mobility of the ion. - [*R*]{.math.inline} is the gas constant. - [*T*]{.math.inline} is the absolute temperature (in Kelvin). - [*F*]{.math.inline} is the Faraday constant. - [*l*]{.math.inline} is the membrane thickness. - [*β*]{.math.inline} is the partition coefficient. Effects on permeability: - [ *β*↑ ]{.math.inline} (increase in the partition coefficient) → [ *P*↑ ]{.math.inline} (permeability increases). - [ *l*↑ ]{.math.inline} (increase in membrane thickness) → [ *P*↓ ]{.math.inline} (permeability decreases). Goldman-Hodgkin-Katz (GHK) Current Equation The **GHK current equation** describes the ionic current ([*I*~*c*~]{.math.inline}) across a membrane assuming a constant electric field. It is given by: \ [\$\$I\_{c} = P\_{c}z\_{c}F\\left( \\frac{z\_{c}\\text{VF}}{\\text{RT}} \\right)\\frac{\\lbrack C\\rbrack\_{\\mathrm{\\text{in}}} - \\lbrack C\\rbrack\_{\\mathrm{\\text{out}}}e\^{\\left( \\frac{- z\_{c}\\text{VF}}{\\text{RT}} \\right)}}{1 - e\^{\\left( \\frac{- z\_{c}\\text{VF}}{\\text{RT}} \\right)}}\$\$]{.math.display}\ Where: - [*I*~*c*~]{.math.inline} is the current due to the ion species. - [*P*~*c*~]{.math.inline} is the permeability of the membrane for the ion. - [*z*~*c*~]{.math.inline} is the valence (charge) of the ion. - [*F*]{.math.inline} is the Faraday constant. - [*V*]{.math.inline} is the membrane potential. - [*R*]{.math.inline} is the gas constant. - [*T*]{.math.inline} is the absolute temperature. - [\[*C*\]~in~]{.math.inline} and [\[*C*\]~out~]{.math.inline} are the ion concentrations inside and outside the membrane. Inward and Outward Current Components The total current can be broken down into inward and outward components. **Inward Current (**[*I*~in~]{.math.inline}**):** \ [\$\$I\_{\\mathrm{\\text{in}}} = - P\_{c}z\_{c}F\\left( \\frac{z\_{c}\\text{VF}}{\\text{RT}} \\right)\\frac{\\lbrack C\\rbrack\_{\\mathrm{\\text{out}}}e\^{\\left( \\frac{z\_{c}\\text{VF}}{\\text{RT}} \\right)}}{1 - e\^{\\left( \\frac{z\_{c}\\text{VF}}{\\text{RT}} \\right)}}\$\$]{.math.display}\ This equation represents the current caused by ions moving into the cell, driven by external ion concentrations. **Outward Current (**[*I*~out~]{.math.inline}**):** \ [\$\$I\_{\\mathrm{\\text{out}}} = P\_{c}z\_{c}F\\left( \\frac{z\_{c}\\text{VF}}{\\text{RT}} \\right)\\frac{\\lbrack C\\rbrack\_{\\mathrm{\\text{in}}}}{1 - e\^{\\left( \\frac{- z\_{c}\\text{VF}}{\\text{RT}} \\right)}}\$\$]{.math.display}\ This equation models the current caused by ions moving out of the cell, driven by internal ion concentrations. Rectification - I ↓ (current/flux) against concentration gradient (see blue line for negative current in Figure) - For + charged ion: - If [ \[*C*\]~out~↓ ]{.math.inline} than [\[*C*\]~in~]{.math.inline} -\> [*E*~rev~]{.math.inline} ↓ (more negative, hyperpolarizing; see Figure below) - If [ \[*C*\]~out~↑  *than* \[*C*\]~in~ ]{.math.inline}-\> [*E*~rev~]{.math.inline} ↑ (more positive, depolarizing)  Goldman-Hodgkin-Katz (GHK) Voltage Equation At rest, the total net current equals zero: \ [∑*I*~total~ = *I*~*K*~ + *I*~Na~ + *I*~Cl~]{.math.display}\ Where each [*I*~*c*~]{.math.inline} follows the GHK current equation for the respective ion species. Therefore, at rest: \ [∑*I* = 0 = *I*~*K*~ + *I*~Na~ + *I*~Cl~]{.math.display}\ The resting membrane potential ([*V*~rest~]{.math.inline}) is: \ [\$\$V\_{\\mathrm{\\text{rest}}} = \\frac{\\text{RT}}{F}\\ln\\left( \\frac{P\_{K}\\lbrack K\\rbrack\_{\\mathrm{\\text{out}}} + P\_{\\mathrm{\\text{Na}}}\\lbrack\\mathrm{\\text{Na}}\\rbrack\_{\\mathrm{\\text{out}}} + P\_{\\mathrm{\\text{Cl}}}\\lbrack\\mathrm{\\text{Cl}}\\rbrack\_{\\mathrm{\\text{in}}}}{P\_{K}\\lbrack K\\rbrack\_{\\mathrm{\\text{in}}} + P\_{\\mathrm{\\text{Na}}}\\lbrack\\mathrm{\\text{Na}}\\rbrack\_{\\mathrm{\\text{in}}} + P\_{\\mathrm{\\text{Cl}}}\\lbrack\\mathrm{\\text{Cl}}\\rbrack\_{\\mathrm{\\text{out}}}} \\right)\$\$]{.math.display}\ IV Curve for a Neuron - **Positive current**: The extracellular environment becomes more positive compared to the baseline. - **Negative current**: The extracellular environment becomes more negative compared to the baseline. The rate of change of voltage over time is given by: \ [\$\$\\frac{\\text{dV}}{\\text{dt}} = - I\\left( V \\right)\$\$]{.math.display}\ At equilibrium ([*I* = 0]{.math.inline}): \ [\$\$\\frac{\\text{dV}}{\\text{dt}} = 0\$\$]{.math.display}\ Stable (blue) & unstable equilibrium (orange) (see Figure) Equivalent Circuits - **IV curve of a resistor**: \ [\$\$I = \\frac{\\text{ΔV}}{R} = \\text{ΔV} \\cdot g = \\left( V - V\_{\\mathrm{\\text{rev}}} \\right) \\cdot g\$\$]{.math.display}\ Where [*g*]{.math.inline} is the conductance, and [*V*~rev~]{.math.inline} is the reversal potential. - **Capacitor**: \ [\$\$I = C \\cdot \\frac{\\text{dV}}{\\text{dt}}\$\$]{.math.display}\ Resistor vs. Capacitor Property Resistor Capacitor --------------------- ---------------------- ---------------------------------- Movement of charges Charges move through Charges accumulate on both sides Parallel R [↓]{.math.inline} C [↑]{.math.inline} Series R[ ↑]{.math.inline} C [↓]{.math.inline} Stages of Membrane Potential *Stage 0: Initial Condition (t = 0)* - **Condition**: No current applied ([*I*~*m*~ = 0]{.math.inline}). - **Membrane potential**: [*V*~*m*~ = *E*~*R*~]{.math.inline} (no voltage change). - **Rate of change**: \ [\$\$\\frac{dV\_{m}}{\\text{dt}} = 0\$\$]{.math.display}\ - **In summary**: \ [\$\$\\left. \\ I\_{m} = 0\\quad \\Rightarrow \\quad V\_{m} = E\_{R}\\quad \\Rightarrow \\quad g\_{m}\\left( V\_{m} - E\_{R} \\right) = 0\\quad \\Rightarrow \\quad\\frac{dV\_{m}}{\\text{dt}} = 0 \\right.\\ \$\$]{.math.display}\ *Stage 1: Intermediate Step (t \> 0)* - **Condition**: Current [*I*~*m*~]{.math.inline} is applied. - **Membrane potential**: [*V*~*m*~ \> *E*~*R*~]{.math.inline} (increases from rest). - **Rate of change**: \ [\$\$\\frac{dV\_{m}}{\\text{dt}} = \\frac{I\_{m}}{C\_{m}} - \\frac{g\_{m}\\left( V\_{m} - E\_{R} \\right)}{C\_{m}}\$\$]{.math.display}\ - **In summary**: \ [\$\$\\left. \\ I\_{m} \> 0 \\Rightarrow V\_{m}\\mathrm{\\text{\~increases}} \> E\_{R} \\Rightarrow g\_{m}\\left( V\_{m} - E\_{R} \\right) \\uparrow \> 0 \\Rightarrow \\frac{dV\_{m}}{\\text{dt}} \\downarrow \\left( \\mathrm{\\text{slower\~voltage\~increase}} \\right) \\right.\\ \$\$]{.math.display}\ *Stage 2: Steady-State (t → ∞)* - **Condition**: Current flow stabilizes. - **Membrane potential**: \ [\$\$V\_{m} = E\_{R} + \\frac{I\_{m}}{g\_{m}}\\left( \\mathrm{\\text{constant}} \\right).\$\$]{.math.display}\ - **Rate of change**: \ [\$\$\\frac{dV\_{m}}{\\text{dt}} = 0\$\$]{.math.display}\ - **In summary**: [\$\\left. \\ I\_{m}\\mathrm{\\ (fully\\ balanced)} \\Rightarrow V\_{m}\\mathrm{\\text{\~constant}} \> E\_{R} \\Rightarrow g\_{m}\\left( V\_{m}{- E}\_{R} \\right)\\mathrm{\\text{\~at\~largest\~change\~and\~constant}} \\Rightarrow \\frac{dV\_{m}}{\\text{dt}} = 0\\ \\left( \\mathrm{\\text{no\~voltage\~increase}} \\right) \\right.\\ \$]{.math.inline} Membrane Potential Over Time The membrane potential over time is given by: \ [\$\$V\_{m} = E\_{R} + \\frac{I\_{m}}{g\_{m}} \\cdot \\left( 1 - e\^{\\left( \\frac{- t}{\\tau} \\right)} \\right)\$\$]{.math.display}\ or \ [\$\$\\Delta V\_{m}\\left( t \\right) = I\_{m} \\cdot R\_{m} \\cdot \\left( 1 - e\^{\\left( \\frac{- t}{\\tau} \\right)} \\right)\$\$]{.math.display}\ Where [\$\\tau = R\_{m} \\cdot C\_{m} = \\frac{R\_{m}}{g\_{m}}\$]{.math.inline} is the time constant. (Maybe could add some of the other equations too but we did not discuss them in depth) ### Lecture 3 Isopotential Sphere vs Cylinder - **Isopotential Sphere**: Voltage is uniform across the membrane. - **Cylinder**: Voltage varies along the long axis of the membrane. Cell Membrane: Resistance & Capacitance - Membrane = parallel resistance (ion channels) + capacitance (membrane itself). - **Higher capacitance** (↑) → less current (↓) flows through ion channels. Cable Model -- Assumptions 1. Membrane parameters [*r*~*m*~]{.math.inline}, [*c*~*m*~]{.math.inline}, [*r*~*a*~]{.math.inline} are linear and uniform throughout. 2. Current flow: - Axial and membrane current. - Radial current = 0. 3. Extracellular resistance [*r*~*o*~ = 0]{.math.inline}. 4. Constant diameter. Cylinder Parameters 1. **Axial/internal resistance** [*r*~*a*~]{.math.inline} or [*r*~*i*~]{.math.inline} (Ω/cm). 2. **Membrane resistance** [*r*~*m*~]{.math.inline} (Ω·cm). 3. **Membrane capacitance** [*c*~*m*~]{.math.inline} (F/cm). 4. **Membrane current** [*i*~*m*~]{.math.inline}. 5. **Axial current** [*i*~*a*~]{.math.inline}. Geometry-Independent Parameters 1. **Specific intracellular resistivity** [*R*~*i*~]{.math.inline} (Ω·cm). 2. **Specific membrane resistivity** [*R*~*m*~]{.math.inline} (Ω·cm²). 3. **Specific membrane capacitance** [*C*~*m*~]{.math.inline} (F/cm²). Cable Equation 1. Axial current over distance: \ [\$\$- \\frac{di\_{a}}{\\text{dx}} = i\_{m}\$\$]{.math.display}\ 2. Voltage variation along membrane: \ [\$\$\\frac{dV\_{m}}{\\text{dx}} = - i\_{a} \\cdot r\_{a}\$\$]{.math.display}\ 3. Full cable equation: \ [\$\$\\frac{1}{r\_{a}}\\frac{d\^{2}V\_{m}}{dx\^{2}} = c\_{m}\\frac{dV\_{m}}{\\text{dt}} + \\frac{V\_{m}}{r\_{m}}\$\$]{.math.display}\ Length & Time Constants - **Space or length constant** ([*λ*]{.math.inline}) (Distance needed to reach 37% of maximum voltage change from the resting potential): \ [\$\$\\lambda = \\sqrt{\\frac{r\_{m}}{r\_{a}}}\$\$]{.math.display}\ - **Time constant** ([*τ*]{.math.inline}) (Time needed to reach 63% of maximum voltage change from the resting potential): \ [*τ* = *r*~*m*~ ⋅ *c*~*m*~]{.math.display}\ - Voltage as a function of distance: \ [\$\$V\\left( x \\right) = V\_{0}e\^{\\left( - \\frac{x}{\\lambda} \\right)}\$\$]{.math.display}\ Conduction Velocity - **Conduction velocity** ([*θ*]{.math.inline}): \ [\$\$\\theta = \\frac{2\\lambda}{\\tau\_{m}}\$\$]{.math.display}\ Effects of Dimension on Parameters - **Axial resistance** ([*r*~*a*~]{.math.inline}): - Diameter ↓ → [*r*~*a*~]{.math.inline} or [*r*~*i*~]{.math.inline} (↑). - Diameter ↑ → [*r*~*a*~]{.math.inline} or [*r*~*i*~]{.math.inline} (↓). - **Membrane resistance** ([*r*~*m*~]{.math.inline}): - ion channels ↓ → [*r*~*m*~]{.math.inline} (↑). - ion channels ↑ → [*r*~*m*~]{.math.inline} (↓). - **Membrane capacitance** ([*c*~*m*~]{.math.inline}): - More membrane → [*c*~*m*~]{.math.inline} (↑). - Less membrane → [*c*~*m*~]{.math.inline} (↓). - Thicker membrane → [*c*~*m*~]{.math.inline} (↓). - Thinner membrane → [*c*~*m*~]{.math.inline} (↑). Current in Extracellular Space - **Current (**[*I*]{.math.inline}**) flows** in extracellular space → small but non-zero resistance [*R*]{.math.inline} in extracellular fluid → small potential difference [*V*]{.math.inline} generated. - **Current sink**: Electric potential ([*V*~near~]{.math.inline}, closer to the channel) becomes slightly more negative (↓) than the surrounding potential ([*V*~far~]{.math.inline}) because less potential is found close to the membrane. - **Current source**: Electric potential ([*V*~near~]{.math.inline}, closer to the channel) becomes slightly more positive (↑) than the surrounding potential ([*V*~far~]{.math.inline}) because more potential is found close to the membrane. -  Signal Range - Signal: tens of μV to a few mV. - Waveform depends on cell type, morphology, and recording location. Extracellular Recordings - **Single electrodes**: Glass pipettes, microwire (Tungsten). - **Multiple electrodes**: Tetrodes, silicon probes (Neuropixels). - **Proximity**: Juxtacellular, loose-patch, cell-attached. Spike Sorting - **Multiple neurons detected** simultaneously → sorting based on spike shape. - **Multiple electrodes** aid in better spike sorting. Intracellular Recordings - **Sharp-electrodes**: Higher resistance (150--180 MΩ), prone to leakage and drift. - **Whole-cell patch-clamp**: Lower resistance (5--20 MΩ), stable. - **In vivo two-photon patching**: Makes cells fluoresce upon pipette contact. **Sharp vs Patch-clamp**: - **Similar**: Firing rates, slow potential transients (e.g., synaptic potentials). - **Different**: Sharp electrodes attenuate fast responses (e.g., action potentials). Voltage Clamp Technique - **Two electrodes**: a. **Recording electrode** measures membrane voltage ([*V*~*m*~]{.math.inline}). b. **Stimulation electrode** brings the membrane to a specific voltage. - Inject or withdraw current, first creating a capacitive current to charge the membrane. - Then observe how current unfolds. Voltage Clamp vs Current Clamp - **Voltage clamp**: Controls voltage, measures current. - **Current clamp**: Controls current, measures voltage. - **Conductance clamp**: Imitates synaptic or voltage-dependent conductance. Single-Electrode Voltage Clamp - For situations where two electrodes can't be used, a patch clamp amplifier with feedback resistor is used. Patch-Clamp Configurations & Use Cases 1. **Cell-attached**: Intact cell, single-channel or capacitive current recordings. 2. **Whole-cell**: Intracellular recordings. 3. **Outside-out**: Access extracellular domain, single-channel recordings. 4. **Inside-out**: Access intracellular domain, single-channel recordings. Single-Channel Current Recordings - Patch-clamp voltage clamp measures single-channel events. - Holding cells at different potentials → observe amplitude and direction of channel openings. Voltage-Dependence of Ion Channels - Ion channels show diversity in voltage-dependence. - Can be studied via single-channel recordings and use of toxins. Non-Electrophysiological Techniques - **Voltage-sensitive dyes** and **calcium imaging**: Used for measuring membrane potential or ion concentration changes. ### Lecture 4 Structure-Function Relationship - **Ion channels**: - **Ion selectivity** (permeation). - **Gating** (how they open). - **Voltage sensing/activation**. - **Inactivation**. Ion Channels vs Transporters - **Ion channels**: Open pores allowing ions to move down electrochemical gradients. - **Transporters**: Actively transport ions against gradients. - **Primary active transport**: Direct use of ATP for transport (e.g., Na+/K+ ATPase). - **Secondary active transport**: Uses electrochemical gradient set by primary transport (e.g., cotransporters). - **Cotransporter types**: - **Antiporter**: Ions move in opposite directions. - **Symporter**: Ions move in the same direction. Glutamate Transporter - **Facilitated transport**: Conformational changes allow ion movement, but no contact between intracellular and extracellular spaces simultaneously. Three Families of Ion Channels 5. **Ligand-gated ion channels**. 6. **Gap junctions**. 7. **Voltage-gated ion channels**. Distinguishing Between Ion Channels 8. **Selectivity**: Which ions pass? 9. **Conductance**: How easily do ions pass? 10. **Gating**: How are they opened? Voltage-Gated K+ Channels - **Tetrameric structure**: 4 subunits. - **Functional parts**: - **S1-S4**: Voltage sensing (S4 particularly). - **P region**: Ion selectivity. - **S5-S6**: Gating and pore. - **Intracellular loops**: Inactivation. Types of K+ Channels 11. **Voltage-gated Kv channels** (40 genes, Kv1-12). 12. **Ca2+-activated (KCa) channels** (5 genes). 13. **Two-pore (K2P) channels** (15 genes). 14. **Inward-rectifying (KIR) channels** (15 genes). Ion Selectivity of K+ Channels - **K+ permeability**: 10,000x more permeant than Na+. - **KcsA channel structure**: - **Bacterial channel**, tetramer. - Each subunit: 2 transmembrane helices (inner and outer), 1 pore helix. - Central water-filled cavity. - **Alpha subunit** is necessary to stabilize the structure and facilitate function. K+ Selectivity Filter Mechanism  **K+ ion diffusion**: - **K+ (larger, 1.33 Å)**: Has \~8 water molecules surrounding it -\> still easier to dehydrate -\> enters the channel hydrated → moves into the central cavity (still surrounded by water molecules) → enters the selectivity filter where 8 carbonyl oxygens (2 bonds at each of the 4 subunits) replace the hydration shell → K+ becomes fully dehydrated → passes smoothly through the filter due to a precise fit and low energy cost. **Na+ exclusion**: - **Na+ (smaller, 0.95 Å)**: Stronger interaction with 3 water molecules or anionic sites (when dehydrated) → difficult to dehydrate → doesn't fit well in the filter\'s 8 oxygen coordination sites → Na+ gets stuck, remains partially hydrated, or becomes dehydrated but still stuck between the oxygens → energetically unfavorable to pass through the filter **I understand it but discuss binding sites once again!** **Previously thought mechanism:** Water molecules (partially dehydrated) between K+ ions when K+ ions move along the channel**.** **Revised mechanism:** No water molecules (fully dehydrated) between K+ ions when K+ ions move along the channel**.** Voltage-Gated Na+ Channels - **Tetrameric structure**: 4 subunits. - **Functional parts**: - **S1-S4**: Voltage sensing (especially S4). - **P region**: Ion selectivity. - **S5-S6**: Gating and pore. - **Inactivation gate**: Controls Na+ channel inactivation. Na+ Selectivity Filter Mechanism - **High field strength** in the center of the Na+ channel\'s selectivity filter creates a strong electric field that attracts Na+ ions, ensuring selective entry. Selective Binding Sites - **K+ channels**: Parallel arrangement of binding sites. - **Cl- channels**: Antiparallel arrangement of binding sites. Conductance and Gating Currents - **Conductance**: - **As membrane potential becomes more positive** (intracellular ↑, depolarization) → positively charged domain (S4) moves upwards (↑) → repelled by positive charges inside the neuron. - **Gating current**: Movement of this domain redistributes charges, creating a measurable current. Gating of K+ Channels - **S4 linker**: Positioned between S5 (gating) and is responsible for voltage-gated K+ channel activation. TTX Blockade - **Tetrodotoxin (TTX)**: Blocks sodium channels by binding to the pore, inhibiting function. Functional Importance of the Alpha Subunit - **Alpha subunit**: Essential for functional ion channels, stabilizes the structure and enables proper gating and ion selectivity. ### Lecture 5 Epilepsy Overview - **Types**: - **Focal onset**: Aware or impaired awareness, can evolve into bilateral tonic-clonic. - **Generalized onset**: Includes tonic-clonic, absence, etc. - **Unknown onset**: Motor or non-motor. - **Seizure Phases**: - **Tonic phase**: Plateau depolarization, rapid firing. - **Clonic phase**: Rhythmic bursts of action potentials. Causes of Epileptic Seizures - **Abnormal excitability**: Neurons generate action potentials too easily. **How excitability becomes abnormal**: - **Channelopathies**: Mutations in ion channels (e.g., Na+, K+). - **Neuronal network imbalance**: Abnormal balance between excitation and inhibition. - **Epilepsy genes**: - 17% voltage-gated ion channels. - 10% ligand-gated ion channels. - 73% other genes. Dravet Syndrome - **Characteristics**: - Rare genetic epileptic encephalopathy. - Begins in the first year of life, often with febrile seizures. - Difficult to treat pharmacologically. - Developmental delay. - **Cause**: - **SCN1A gene** mutation → affects NaV1.1 (sodium channel). - **Decreased sodium channels** in hippocampal interneurons → overexcitation of pyramidal neurons. - **Severity Based on NaV1.1 mutation**: - **Mild mutation**: Febrile seizures. - **Medium mutation**: GEFS+ (generalized epilepsy with febrile seizures plus). - **Severe mutation**: Dravet syndrome (SMEI, severe myoclonic epilepsy of infancy). Channelopathies and Expected Changes **Channelopathy** **Depolarization** **Hyperpolarization** ------------------- ------------------------------- ---------------------------------- **Na+ channel** ↓ depolarization (↓Na+, ↑ K+) ↑ hyperpolarization (↓Na+, ↑ K+) **K+ channel** ↑ depolarization (↑Na+, ↓K+) ↓hyperpolarization (↑Na+, ↓K+) Epilepsy Treatment Strategies - **General goal**: Restore balance between excitation and inhibition. - **Challenges**: Incomplete understanding of epileptogenesis, risk of interfering with normal synaptic activity. - **Focus**: - Prevent long-lasting depolarization. - Prevent high-frequency, synchronous firing. Epileptic Drugs - **Mechanism**: Most drugs act on multiple points; side effects determine prescriptions. - **Phenytoin/Carbamazepine/Lamotrigine**: - Inhibit Na+ channels by: - **Inhibiting activation** (reducing transition from closed to open). - **Speeding up inactivation**. - **Slowing recovery** from inactivation. - Achieve **use-dependent block** by binding to the activated state. - **Effect**: - No effect on single action potential. - Prolongs the refractory period (inhibits repetitive firing) Absence Epilepsy - **Firing behavior**: Thalamocortical neurons. - **Role of T-type Ca2+ channels**: Involved in burst firing and corticothalamic feedback. ### Lecture 6 Bernstein's Membrane Hypothesis: - Axon: High K+, low Cl-. - At rest: only K+ crosses → Cells are negatively charged. - During action: the membrane breaks down → creating negativity outside. How can we test if an axon is negatively charged? - Record from the inside. - Why hasn\'t someone done that experiment yet? → Axons are small. - Most axons diameter \< 0.2 mm → squid giant axon diameter \~1 mm. Hodgkin and Huxley measured AP in squid giant axon Evaluation of Bernstein's membrane hypothesis: - Rest: mainly K+ crosses → negative potential → ✅ (correct). - Action: membrane breaks down → creating negativity → ❌ (incorrect). - But what explains the positive membrane potential detected by Hodgkin and Huxley? → At action: the inside of the axon becomes positively charged -- how? The Sodium Hypothesis: - Test this hypothesis by changing the sodium equilibrium potential (ENa): a. Manipulate ENa by changing the extracellular concentration → decreasing Na+ decreases action potential amplitude and prolongs it. b. Replace Na+ with dextrose → reduces amplitude → reset. **Sodium ion entering the cell is responsible for the inside of the axon becoming positively charged (depolarization).** **AP shape becomes different. How to study this?** - Cole & Curtis: one-electrode voltage-clamp. - Hodgkin, Huxley & Katz: two-electrode voltage-clamp. Voltage Clamp Experiments: **Positive Current Injection:** - Voltage Difference Calculation: - Amplifier calculates membrane potential [*V*~*m*~]{.math.inline} by subtracting reference voltage [*V*~ref~]{.math.inline} from the input voltage [*V*~in~]{.math.inline}: \ [*V*~*m*~ = *V*~in~ − *V*~ref~]{.math.display}\ - The goal: keep [*V*~*m*~]{.math.inline} at clamp voltage [*V*~clamp~]{.math.inline}. - Positive Current Injection: - When [*V*~clamp~ \> *V*~*m*~]{.math.inline}, system injects positive current [*I*~inj~]{.math.inline} into the axon → raises [*V*~*m*~]{.math.inline} to desired level. - Red arrows show positive current injected. - Maintaining the Clamp: - System adjusts the injected current as necessary to maintain [*V*~*m*~]{.math.inline} at [*V*~clamp~]{.math.inline}. **Second Figure (Negative Current Injection):** - Voltage Difference Calculation: - Same as positive injection figure. - Negative Current Injection: - When [*V*~clamp~ \